Recent Advances in Mathematical Modeling, Numerical Analysis, and Interdisciplinary Applications

1. Introduction

Mathematical modeling and numerical analysis have become a common language for describing systems whose behavior cannot be captured by observation alone. In modern applications, the model is no longer only a compact representation of a known physical law; it is also a tool for hypothesis testing, design optimization, parameter identification, image interpretation, uncertainty assessment, and decision support. This is particularly evident when the object of study involves moving boundaries, many interacting degrees of freedom, heterogeneous populations, scarce experimental data, biomedical coupling, or thermodynamic sensitivity. The Special Issue “Mathematical Modeling and Numerical Analysis with Applications in Various Fields” was organized around this broad methodological perspective. The twelve papers collected in it demonstrate how analytical derivations, computational models, reduced-order techniques, stochastic processes, computer vision, and sensitivity analysis can be combined to address applied problems across geometry processing, phase transitions, structural dynamics, mechanochemistry, multiphase flows, biomedical mechanics, anomalous transport, energy systems, and sustainability modeling.

A distinctive feature of this collection is that the papers do not merely apply existing algorithms to isolated case studies. Instead, they expose the modeling choices that must be made when a real system is simplified without losing its essential behavior. In some papers, the central challenge is geometric: how to reconstruct or preserve a surface while maintaining mathematically meaningful constraints. In others, the difficulty lies in the dynamics of a moving interface, the cost of high-dimensional simulations, the interpretation of synthetic and experimental image data, or the sensitivity of engineering predictions to constitutive or thermodynamic assumptions. Taken together, the papers show that successful applied mathematics depends on the interplay between fidelity and tractability, and that analytical insight and numerical computation are most powerful when developed in close connection with the physical or technological context.

2. Overview of the Special Issue

2.1. Geometry, Interfaces and Phase-Transition Models

The collection begins naturally with the problem of shape and geometry. Kouibia, Pasadas, and Omri addressed a shape-preserving variational spline approximation problem for filling holes in generalized offset surfaces [1]. Their study belongs to computer-aided geometric design, but its implications are wider: many numerical models begin with incomplete, noisy, or partially reconstructed geometries, and the quality of subsequent simulation depends on whether the reconstructed surface preserves relevant shape features. By proving existence and uniqueness properties and analyzing computation and convergence, this contribution illustrates how approximation theory can serve as a rigorous foundation for geometric completion problems that arise before a physical or engineering model is even solved.

Once geometry becomes dynamic, the mathematical structure changes from reconstruction to evolution. This is evident in the study by Alexandrov, Alexandrova, Ivanov, and Toropova on directional crystallization of binary liquids [2]. Instead of relying on the classical simplification of a macroscopically planar phase interface, the authors examined the role of a two-phase mushy region and replaced the full moving mushy layer by an equivalent discontinuity-interface model that preserves its main physical influence. Their analytical solutions for steady-state and self-similar regimes show that the mushy region can alter solute concentrations by several tens of percent relative to the planar-interface approximation. This result is important not only for crystallization theory, but also for applied modeling more generally: it demonstrates that apparently small changes in interface representation can lead to large changes in physically relevant predictions.

A related moving-boundary problem appears in the paper by Nizovtseva and Ankudinov on underwater ice evolution and nonlinear dynamics of false bottoms [3]. False bottoms form in a coupled thermal and solutal environment, where ice solidification, salinity redistribution, and mushy-zone dynamics interact. The authors compared an analytical nonlinear model solved in integral form with a binary phase-field formulation for an aqueous NaCl solution. The agreement between the two descriptions emphasizes the complementary roles of analytical and numerical modeling: analytical integral solutions provide structure, interpretation and benchmarks, whereas phase-field simulations make it possible to follow coupled heat and solute transfer in a spatially resolved numerical framework. Together with the crystallization study [2], this paper shows how moving-boundary and phase-transition problems benefit from the joint development of reduced analytical representations and fully numerical descriptions.

2.2. Reduced-Order Modeling, Structural Dynamics and Digital Twins

A second group of papers addresses the computational cost of dynamic analysis. In structural mechanics, high-fidelity models are increasingly expected to operate not only as offline simulations but also as components of digital twins, monitoring systems, and iterative design loops. Choi, Lee, and Chang proposed a dynamic-condensation-based reduction method for precise broadband frequency analysis [4]. Conventional dynamic condensation can lose accuracy in higher-frequency ranges when a single reduced model is used throughout the analysis. The proposed degree-of-freedom-based adaptive strategy constructs frequency-aware reduced models and uses a Taylor-series formulation to obtain a frequency-independent transformation matrix suitable for interpolation. This approach directly addresses a recurring challenge in numerical analysis: how to retain the essential behavior of a large model while reducing the repeated computational burden associated with frequency response calculations.

The same need for efficient but reliable mechanics-based prediction is central to the frequency-based finite element updating method for physics-based digital twins developed by Jeon, Choi, Ahn, Lee, and Chang [5]. Their work recognizes that a useful digital twin must reproduce the behavior of an actual structure with sufficient accuracy while remaining computationally feasible. Instead of relying on time-domain updating alone, the authors formulate a frequency-based approach that incorporates stiffness, mass and damping effects and combines finite element updating with model reduction. Their numerical example involving a control rod drive mechanism demonstrates that substantial reductions in computation time can be achieved while maintaining accuracy. Read together, the two structural-dynamics papers [4,5] show how reduced-order modeling is becoming a core requirement for practical digital twins, especially when broadband dynamic behavior, vibration analysis, and repeated model updates are required.

2.3. Multiphysics Processes, Data Generation and Energy-System Sensitivity and Sustainability Modeling

Several papers in the Special Issue extend mathematical modeling into strongly coupled multiphysics and data-assisted settings. Son developed a lumped-parameter mathematical model for a high-energy shaker mill process with one-dimensional oscillatory ball motion and collisional heat generation [6]. The model combines Hamiltonian mechanics, thermomechanical coupling, collisional heat production and heat exchange among the milling ball, vial and surrounding air. By using a lumped-parameter representation, the study converts a complex mechanochemical environment into a computationally efficient model that can be validated against temperature data and used to anticipate heat production under different operating conditions. This paper is a clear example of how an applied mathematical model can support process understanding and optimization when direct measurement of all energy-transfer pathways is difficult.

The contribution by Mikushin, Martynenko, Nizovtseva, Makhaeva, Nikishina, Chernushkin, Lezhnin, and Starodumov connects mathematical modeling with computer vision and synthetic-data generation for multiphase systems [7]. Bubble flows are important in biotechnology, medicine, water treatment, oil and gas, and other areas, but the training of neural-network-based recognition tools is often limited by the cost of manually annotated data. The authors used Superformula regression to model bubble boundaries and to generate artificial bubble images under physically meaningful conditions. This work shows that mathematical parametrization can play a constructive role in data-driven modeling: instead of treating machine learning as a replacement for physical modeling, it uses a shape model to create labeled datasets and improve boundary detection for experimental and computational analysis.

Modeling choices also have direct consequences in energy conversion. Alharbi and Alshammari investigated the sensitivity of Organic Rankine Cycle performance predictions to the choice of equation of state [8]. Their comparison of ideal-gas, cubic, and more advanced real-fluid formulations demonstrates that thermodynamic-property models are not merely technical details; they affect predicted cycle efficiency and net power output. The study is therefore relevant beyond the specific working fluid and cycle considered. It reminds modelers that the reliability of numerical results depends on the consistency of the constitutive and property models embedded in the computation.

At a broader energy and environmental scale, Wang, Xu, Song, Sheng, and Wang constructed a dynamic model integrating technological progress, carbon emissions, economic growth, and energy structure [9]. This contribution moves the Special Issue from component-level energy-conversion modeling to sustainability-oriented system modeling, where the governing relationships are not only thermodynamic but also socio-economic. By coupling technological change with energy-structure optimization, emission behavior, and growth dynamics, the study highlights another role of mathematical modeling: to organize interdependent policy-relevant variables into a framework that can be used to analyze low-carbon development pathways.

The shaker-mill, bubble-flow, and energy-system papers [6,7,8], together with the sustainability-oriented dynamic model [9], collectively underline that applied models must be judged not only by mathematical elegance, but also by their ability to represent energy transfer, data limitations, socio-economic coupling, and sensitivity to physical assumptions.

2.4. Biomedical Mechanics and Stochastic Transport Across Scales

Biomedical applications in the Special Issue illustrate how mathematical and numerical models can connect mechanical, particle-scale, and population-level descriptions. Brusokas, Borodinas, and Jasevičius studied the three-dimensional behavior of a blood clot in a vein under the mechanical influence of blood flow [10]. Their simulations compare fixed and flexible vessel walls and include different levels of clot viscoelasticity. The results highlight the dependence of clot deformation, vessel-wall deformation, stresses, and flow velocity on the mechanical properties of the vessel. This work demonstrates the importance of coupling fluid-mechanical and solid-mechanical effects when modeling thrombus behavior, especially when changes in lumen shape and clot orientation may affect detachment risk and flow conditions.

A more particle-level biomedical model is developed by Jasevičius in the numerical study of cholesterol particle interaction with blood-vessel surfaces [11]. The paper considers the ability of individual cholesterol particles to interact with and adhere to a vessel wall under the action of different forces. Although simplified relative to the full biochemical and hemodynamic complexity of atherosclerosis, such particle-based modeling is valuable because it isolates mechanical mechanisms of adhesion and motion that may be obscured in purely continuum descriptions. In this sense, the clot and cholesterol studies [10,11] represent complementary scales of biomedical modeling: one resolves a deformable macroscopic thrombus in a vessel, while the other focuses on the mechanical interaction of an individual particle with the vessel surface.

The probabilistic dimension of transport appears in the work of Fedotov, Ivanov, and Zhang on anomalous transport of a heterogeneous population and a time-changed Pólya process [12]. The authors introduce a continuous-time unidirectional random-walk model with subdiffusive trapping and beta-distributed jump probabilities. This formulation captures heterogeneity at the population level and leads to ensemble self-reinforcement, with the average particle position represented through a time-changed Pólya process involving an inverse stable subordinator. The paper broadens the Special Issue by showing that numerical and analytical modeling of applied phenomena is not limited to deterministic systems. Heterogeneity, memory effects, and anomalous transport often require stochastic structures capable of connecting individual transition mechanisms with ensemble behavior.

2.5. Cross-Cutting Methodological Themes and Outlook

Across these twelve papers, several methodological themes emerge. The first is the importance of selecting the right level of model fidelity. Shape-preserving spline approximation [1], discontinuity-interface crystallization models [2], lumped-parameter thermomechanics [6], reduced-order frequency analysis [4], and frequency-based digital-twin updating [5] all reduce complexity, but none of them do so arbitrarily. In each case, simplification is useful because it preserves the variables, constraints, or mechanisms that are essential for the intended prediction. The Special Issue therefore reinforces a central lesson of applied mathematics: a good model is not necessarily the most detailed model, but the one whose assumptions are transparent, whose limitations are understood, and whose outputs are reliable for the question being asked.

The second theme is the productive combination of analytical, numerical, and data-driven methods. Analytical solutions and integral formulations provide benchmarks and physical interpretation in moving-boundary problems [2,3]. Numerical finite element and dynamic reduction methods enable large structural models to be used in realistic workflows [4,5]. Synthetic image generation and Superformula-based boundary modeling support machine-learning applications where experimental annotation is expensive [7]. Stochastic transport theory supplies a framework for heterogeneity and memory effects [12]. These examples suggest that the future of numerical analysis in applied fields will not be divided into purely analytical, purely computational, or purely data-driven categories. Instead, robust modeling will increasingly depend on hybrid approaches in which each component compensates for the weaknesses of the others.

The third theme is validation and sensitivity. Biomedical models must account for vessel elasticity, clot viscoelasticity, and particle-wall interaction [10,11]; energy-system models must account for the effect of thermodynamic-property assumptions [8]; while sustainability-oriented dynamic models must integrate technological progress, carbon-emission behavior, economic growth, and energy-structure change [9]; and phase-transition models must account for how mushy layers modify heat and mass transfer [2,3]. Future research should therefore place greater emphasis on uncertainty quantification, parameter sensitivity, reproducible benchmark problems, and transparent reporting of model assumptions. Such efforts are especially important when mathematical models are used to support engineering design, biomedical interpretation, environmental prediction, or energy-system optimization.

Looking ahead, promising directions include multiscale models that link particle interactions to continuum fields, phase-field and moving-boundary formulations that remain computationally efficient, real-time reduced-order models for digital twins, and physics-informed data generation methods for experimental diagnostics. The papers in this Special Issue also point toward the value of shared computational benchmarks. Whether the problem is a reconstructed surface, a solidification front, a structural frequency response, a multiphase image dataset, a deformable blood clot, a stochastic transport process, an Organic Rankine Cycle, or a coupled technology–energy–emissions system, progress will be accelerated by models that are not only accurate but also interpretable, reproducible, and transferable.

3. Conclusions

The Special Issue “Mathematical Modeling and Numerical Analysis with Applications in Various Fields” presents a diverse but coherent view of contemporary applied mathematics. Its contributions show that mathematical modeling is most effective when it is connected to the structure of the problem: geometry must be preserved before simulation can be trusted; moving interfaces must be represented with enough fidelity to capture heat and mass transfer; high-dimensional structural models must be reduced without sacrificing accuracy; multiphysics processes require interpretable coupling assumptions; machine-learning workflows benefit from mathematically generated data; biomedical models require careful treatment of mechanical interaction; stochastic transport must account for heterogeneity and memory; and energy-system predictions depend on thermodynamic-property modeling; and sustainability-oriented models must connect technological progress, carbon emissions, economic growth and energy-structure dynamics. The Special Issue therefore demonstrates both the breadth and the unity of mathematical modeling and numerical analysis. It is hoped that the methods and results collected here will encourage further interdisciplinary work in which rigorous mathematical structures, efficient numerical algorithms, and application-specific insight are developed together.